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Theorem pm110.643 8449
Description: 1+1=2 for cardinal number addition, derived from pm54.43 8273 as promised. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 8205), but after applying definitions, our theorem is equivalent. The comment for cdaval 8442 explains why we use  ~~ instead of =. See pm110.643ALT 8450 for a shorter proof that doesn't use pm54.43 8273. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)
Assertion
Ref Expression
pm110.643  |-  ( 1o 
+c  1o )  ~~  2o

Proof of Theorem pm110.643
StepHypRef Expression
1 1on 7029 . . 3  |-  1o  e.  On
2 cdaval 8442 . . 3  |-  ( ( 1o  e.  On  /\  1o  e.  On )  -> 
( 1o  +c  1o )  =  ( ( 1o  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) ) )
31, 1, 2mp2an 672 . 2  |-  ( 1o 
+c  1o )  =  ( ( 1o  X.  { (/) } )  u.  ( 1o  X.  { 1o } ) )
4 xp01disj 7038 . . 3  |-  ( ( 1o  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)
51elexi 3080 . . . . 5  |-  1o  e.  _V
6 0ex 4522 . . . . 5  |-  (/)  e.  _V
75, 6xpsnen 7497 . . . 4  |-  ( 1o 
X.  { (/) } ) 
~~  1o
85, 5xpsnen 7497 . . . 4  |-  ( 1o 
X.  { 1o }
)  ~~  1o
9 pm54.43 8273 . . . 4  |-  ( ( ( 1o  X.  { (/)
} )  ~~  1o  /\  ( 1o  X.  { 1o } )  ~~  1o )  ->  ( ( ( 1o  X.  { (/) } )  i^i  ( 1o 
X.  { 1o }
) )  =  (/)  <->  (
( 1o  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  ~~  2o ) )
107, 8, 9mp2an 672 . . 3  |-  ( ( ( 1o  X.  { (/)
} )  i^i  ( 1o  X.  { 1o }
) )  =  (/)  <->  (
( 1o  X.  { (/)
} )  u.  ( 1o  X.  { 1o }
) )  ~~  2o )
114, 10mpbi 208 . 2  |-  ( ( 1o  X.  { (/) } )  u.  ( 1o 
X.  { 1o }
) )  ~~  2o
123, 11eqbrtri 4411 1  |-  ( 1o 
+c  1o )  ~~  2o
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758    u. cun 3426    i^i cin 3427   (/)c0 3737   {csn 3977   class class class wbr 4392   Oncon0 4819    X. cxp 4938  (class class class)co 6192   1oc1o 7015   2oc2o 7016    ~~ cen 7409    +c ccda 8439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1o 7022  df-2o 7023  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-cda 8440
This theorem is referenced by: (None)
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